NSW Maths (Years 7 to 10)

Multiplying and Dividing Integers: Sign Rules Explained for NSW Year 7–10 Maths

Published 2026-10-26 · Updated 2026-10-26

Many NSW students can add and subtract negative numbers without too much trouble, but multiplying and dividing them is where the wheels fall off. The most common reason is that students try to rely on intuition rather than a clear rule, and intuition fails badly when both numbers are negative. Getting this right matters for your school assessment because sign errors are easy marks to lose on otherwise correct working.

What the syllabus asks

The NESA Mathematics 7–10 curriculum asks you to multiply and divide directed numbers (integers that carry a positive or negative sign). That means you need to know the sign rules for products and quotients, and you need to apply them reliably, including when a chain of multiplications is involved.

The idea, explained

Starting point: multiplication as repeated addition

Before the sign rules, it helps to see where they come from. Consider 4 × (−3). You can read this as four lots of −3 added together:

(−3) + (−3) + (−3) + (−3) = −12

So 4 × (−3) = −12. A positive number multiplied by a negative number gives a negative result. In short: different signs give a negative answer.

When repeated addition runs out of road

Repeated addition works fine for Part A above, but it breaks down when both numbers are negative. You cannot act out "negative two lots of −3" by adding things up. A different argument is needed, and it comes from following a pattern.

Keep the second factor fixed at −3 and step the first factor down one at a time:

3 × (−3) = −9 2 × (−3) = −6 1 × (−3) = −3 0 × (−3) = 0

Notice what happens to the answer at each step: it goes up by 3 every time (−9, −6, −3, 0). There is no reason for that pattern to suddenly stop at zero, so continue it:

(−1) × (−3) = 3 (−2) × (−3) = 6

So (−2) × (−3) = 6, a positive result. Matching signs give a positive answer. You can also check this with a real-world picture: think of −3 as a debt of 3 dollars. Taking away two lots of that debt leaves you 6 dollars better off, which is a positive change.

The two sign rules, stated clearly

• Different signs (one positive, one negative): the answer is negative.
• Same signs (both positive or both negative): the answer is positive.

These rules apply to both multiplication and division. So (−20) ÷ (−4) has matching signs, giving a positive result: 20 ÷ 4 = 5, so (−20) ÷ (−4) = 5.

A reliable three-step method

1. Look at the signs of both numbers. Are they the same or different?
2. Apply the sign rule to find the sign of your answer before you touch the digits.
3. Multiply or divide the digits as if both numbers were positive, then attach the sign from step 2.

Working out the sign separately from the digits is the single most effective habit you can build for this topic.

Chains of three or more factors

When you multiply more than two integers together, apply the sign rule one step at a time, or count the total number of negative signs. An odd number of negative factors always gives a negative result; an even number always gives a positive result. For example, (−2) × (−3) × (−5): the first pair gives +6 (same signs), then 6 × (−5) = −30 (different signs). Three negatives, odd count, negative answer: −30.

What the exam asks

Maths teachers writing school assessment questions on this topic typically ask you to:

• Calculate a product or quotient of two directed numbers, such as (−6) × 5 or (−18) ÷ (−3).
• Work through a chain of multiplications, such as (−2) × (−3) × (−5), where tracking the sign across every step is essential.
• Apply the idea in a worded context, for example: a submarine descends at −15 metres per minute; find its total change in depth after 6 minutes (answer: −90 metres).

In every case, showing your sign reasoning clearly in working is good practice, because maths teachers can award marks for correct method even if a digit error creeps in.

Common mistakes

• Treating (−4)² and −4² as the same expression. They are not. (−4)² means (−4) × (−4) = 16, because the brackets square the whole negative number. −4² means −(4²) = −16, squaring only the 4 and then applying the negative sign. The brackets change what actually gets squared.
• Forgetting to recheck the sign in a chain of factors. A student might correctly find (−2) × (−3) = 6, but then ignore the sign when multiplying by the third factor and write 6 × (−4) = 24 instead of −24. Apply the sign rule at every individual multiplication step.
• Guessing the sign from intuition rather than the rule. Intuition often says "two negatives cancel out" without any clear reasoning. Rely on the rule (same signs positive, different signs negative) and apply it deliberately.
• Mixing up the sign rule with addition rules. When you add two negative numbers you get a more negative result. When you multiply two negative numbers you get a positive result. These are different operations with different rules, and confusing them is a very common source of errors.
• Skipping the sign step and working with digits first. Students who find the numerical value first and then try to remember the sign at the end frequently attach the wrong sign. Always settle the sign before the digits.

If you want structured practice on exactly this material, Avocado's AI-native tutoring service walks you through each step with immediate feedback. Try the Multiplying and Dividing Integers lesson to build the fluency your school assessment requires.